Inspired by 2024 Fall LMT Guts

by sqing, Apr 24, 2025, 12:24 PM

Let $x$ , $y$ , $z$ are pairwise distinct real numbers satisfying $x^2+y =y^2 +z = z^2+x.$ Prove that
$(x+y)(y+z)(z+x)=-1$ Let $x$ , $y$ , $z$ are pairwise distinct real numbers satisfying $x^2+2y =y^2 +2z = z^2+2x.$ Prove that
$(x+y)(y+z)(z+x)=-8$

This post has been edited 2 times. Last edited by sqing, 32 minutes ago

algebra

Solve All 6 IMO 2024 Problems (42/42), New Framework Looking for Feedback

by Blackhole.LightKing, Apr 24, 2025, 12:14 PM

Hi everyone,

I’ve been experimenting with a different way of approaching mathematical problem solving — a framework that emphasizes recursive structures and symbolic alignment rather than conventional step-by-step strategies.

Using this method, I recently attempted all six problems from IMO 2024 and was able to arrive at what I believe are valid full-mark solutions across the board (42/42 total score, by standard grading).

However, I don’t come from a formal competition background, so I’m sure there are gaps in clarity, communication, or even logic that I’m not fully aware of.

If anyone here is willing to take a look and provide feedback, I’d appreciate it — especially regarding:

The correctness and completeness of the proofs

Suggestions on how to make the ideas clearer or more elegant

Whether this approach has any broader potential or known parallels

I'm here to learn more and improve the presentation and thinking behind the work.

You can download the Solution here.

https://agi-origin.com/assets/pdf/AGI-Origin_IMO_2024_Solution.pdf

Thanks in advance,
— BlackholeLight0

Inspired by SXTX (4)2025 Q712

by sqing, Apr 24, 2025, 11:59 AM

Let $a ,b,c>0$ and $(a+b)^2+2(b+c)^2+(c+a)^2=12.$ Prove that $abc(a+b+c) \leq \frac{9}{5}$ Let $a ,b,c>0$ and $2(a+b)^2+ (b+c)^2+2(c+a)^2=12.$ Prove that $abc(a+b+c) \leq \frac{9}{8}$

inequalities proposed

algebra

Find the minimum

by sqing, Jul 25, 2023, 12:14 AM

In acute triangle $ABC$ , Find the minimum of $2\tan A +9\tan B +17 \tan C .$
h h
In acute triangle $ABC$ , Find the minimum of $4\tan A +7\tan B +14 \tan C .$
In acute triangle $ABC$ . Prove that $2\tan A +9\tan B +17 \tan C \geq 40$

Attachments:

This post has been edited 2 times. Last edited by sqing, Jul 27, 2023, 1:28 AM

How many non-attacking pawns can be placed on a $n \times n$ chessboard?

by DylanN, Jan 18, 2021, 1:26 AM

A pawn is a chess piece which attacks the two squares diagonally in front if it. What is the maximum number of pawns which can be placed on an $n \times n$ chessboard such that no two pawns attack each other?

combinatorics

Russian NT with a Ceiling

by naman12, Sep 22, 2020, 11:28 PM

Let $a$ and $b$ be two positive integers. Prove that the integer
$a^2+\left\lceil\frac{4a^2}b\right\rceil$ is not a square. (Here $\lceil z\rceil$ denotes the least integer greater than or equal to $z$ .)

Russia

This post has been edited 2 times. Last edited by naman12, Sep 22, 2020, 11:29 PM

Excircle Tangency Points Concyclic with A

by tastymath75025, Jan 21, 2019, 5:00 PM

Let $ABC$ be a triangle with incenter $I$ , and let $D$ be a point on line $BC$ satisfying $\angle AID=90^{\circ}$ . Let the excircle of triangle $ABC$ opposite the vertex $A$ be tangent to $\overline{BC}$ at $A_1$ . Define points $B_1$ on $\overline{CA}$ and $C_1$ on $\overline{AB}$ analogously, using the excircles opposite $B$ and $C$ , respectively.

Prove that if quadrilateral $AB_1A_1C_1$ is cyclic, then $\overline{AD}$ is tangent to the circumcircle of $\triangle DB_1C_1$ .

Ankan Bhattacharya

This post has been edited 1 time. Last edited by tastymath75025, Jan 21, 2019, 5:08 PM

geometry

Show that XD and AM meet on Gamma

by MathStudent2002, Jul 19, 2017, 4:32 PM

Let $ABC$ be a triangle with circumcircle $\Gamma$ and incenter $I$ and let $M$ be the midpoint of $\overline{BC}$ . The points $D$ , $E$ , $F$ are selected on sides $\overline{BC}$ , $\overline{CA}$ , $\overline{AB}$ such that $\overline{ID} \perp \overline{BC}$ , $\overline{IE}\perp \overline{AI}$ , and $\overline{IF}\perp \overline{AI}$ . Suppose that the circumcircle of $\triangle AEF$ intersects $\Gamma$ at a point $X$ other than $A$ . Prove that lines $XD$ and $AM$ meet on $\Gamma$ .

Proposed by Evan Chen, Taiwan

This post has been edited 2 times. Last edited by v_Enhance, May 6, 2019, 1:36 PM

IMO 2011 Problem 6

by liberator, Jul 21, 2015, 4:57 PM

Problem: Let $ABC$ be an acute triangle with circumcircle $\Gamma$ . Let $\ell$ be a tangent line to $\Gamma$ , and let $\ell_a, \ell_b$ and $\ell_c$ be the lines obtained by reflecting $\ell$ in the lines $BC$ , $CA$ and $AB$ , respectively. Show that the circumcircle of the triangle determined by the lines $\ell_a, \ell_b$ and $\ell_c$ is tangent to the circle $\Gamma$ .

Proposed by Japan

My solution

This post has been edited 2 times. Last edited by liberator, Jul 22, 2015, 1:35 PM

AZE JBMO TST

by IstekOlympiadTeam, May 2, 2015, 7:05 PM

Find all non-negative solutions to the equation $2013^x+2014^y=2015^z$

number theory

AZE JBMO TST

TST

Domain swept by a parabola

by Kunihiko_Chikaya, Feb 25, 2015, 12:09 PM

For a positive real number $a$ , consider the following parabola on the coordinate plane.
$C:\ y=ax^2+\frac{1-4a^2}{4a}$
When $a$ ranges over all positive real numbers, draw the domain of the set swept out by $C$ .

Submit

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by bachkieu, Jan 31, 2025, 1:40 AM
hello...
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2024 shout ftw
by Shreyasharma, Feb 19, 2024, 10:28 PM
time flies
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first 2023 shout
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offline.................
by 799786, Dec 27, 2021, 7:08 AM
YOU SHALL NOT PASS! - liberator
by OlympusHero, Aug 16, 2021, 4:10 AM
Nice Blog!
by geometry6, Jul 31, 2021, 1:39 PM
First shout out in 2021
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indeed a pr0 blog
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pr0 blog !!
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Very nice blog !
by Kamran011, Oct 31, 2019, 5:48 PM
1000 VISITS!!!

IN 10 DAYS!!!
by liberator, Aug 23, 2014, 3:47 PM
YOU ARE SO PRO!!!!

Have you been to the IMO? If you have, then on not bragging about it. If you haven't, HOW ARE YOU SO CLEVER??!!?
by jlammy, Aug 23, 2014, 12:51 PM
Cool blog, great geometry problems!
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Extremely good at Geometry

Could you consider joining an Online Math Open Team with me?
by DanielL2000, Aug 22, 2014, 12:40 AM
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by sjaelee, Aug 21, 2014, 10:00 PM
thanks Cyclicduck!
by liberator, Aug 21, 2014, 9:56 PM
Hi, you are very good at geometry!
by Cyclicduck, Aug 21, 2014, 8:29 PM
Posts? What are those, exactly?
by kmw2001, Aug 19, 2014, 8:50 PM
@kmw2001, have 5 posts first.
by bcp123, Aug 19, 2014, 8:19 PM
How do i blog?
by kmw2001, Aug 19, 2014, 5:30 PM
thx bcp123, I guess its because I only have 10 posts, not much to brag on about
by liberator, Aug 15, 2014, 10:26 PM
why no one is writing here?
keep this going. nice collection of geoish things
by bcp123, Aug 15, 2014, 10:14 PM
thx!
by liberator, Aug 14, 2014, 2:56 PM
nice blog
by bcp123, Aug 14, 2014, 2:39 PM